Designing a contoured beam antenna

Designing a contoured beam antenna

Citation for published version (APA):

Herben, M. H. A. J. (1979). Designing a contoured beam antenna. (EUT report. E, Fac. of Electrical Engineering; Vol. 79-E-104). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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by

Department of Electrical Engineering

Eindhoven The Netherlands

DESIGNING A CONTOURED BEAM ANTENNA

by M.H.A.J. Herben TH-Report 79-E-104 ISBN 90-6144-104-8 Eindhoven December 1979

Summary

Acknowledgement

1. General Introduction

2. The coordinate transformation

2.1 Introduction

2.2 Transformation equations 2.3 Calculations

3. Excitation coefficients of the feeds 3.1 Introduction

3.2 Calculation of the excitation coefficients

4. Calculation of the contoured beam antenna pattern

1 2 3 4 4 4 6 7 7 7 10 4.1 Introduction 10

4.2 The symmetrical parabolic reflector antenna 10 4.2.1 The defocused symmetrical parabolic reflector antenna 10

4.2.2 The 3 dB beamwidth 12

4.2.3 Beam deviation factor 12

4.2.4 Relation between the minimum

Flo,

the diameter of a

feed and the aperture distribution 13 4.2.5 The contoured beam antenna pattern 13 4.3 The offset parabolic reflector antenna 14 4.3.1 The defocused offset parabolic reflector antenna 14 4.3.2 Power n of the cosine illumination function 17

4.3.3 The 3 dB beamwidth 18

4.3.4 Beam deviation factor 18

4.3.5 Relation between the maximum subtended angle ~

and the diameter d of a feed a max 19 4.3.6 Phase of the desired total far field EO'(SP ,8 ,$) 19 4.3.7 The contoured beam antenna pattern m m m 19

S. Further investigation of some properties of the contoured beam

antenna 20

5.1 Introduction 20

5.2 Comparison of contoured beam antennas with conventional

reflector antennas 20

5.3 The WARC specifications 21

5.4 Frequency dependence of the radiation pattern 21

5.5 Aperture distribution 22

5.6 Spillover 23

5.6.1 Spillover of a defocused parabolic reflector antenna 23 5.6.2 Spillover of a contoured beam antenna 24

5.7 Beam efficiency 24 5.8 Gain 26 6. Conclusion References Appendix A Figures 27 28 30 31

Summary

This report describes the design of a contoured beam reflector antenna. The region on earth to be illuminated is transformed to a spherical coor-dinate system with a geostationary satellite as origin. The radiation pattern of a defocused paraboloid (symmetrical as well as offset confi-guration) has been studied.

The important magnitudes required for the design 9

3dB, BDF and are determined. A method has been developed of finding suitable

(F/D) ,

m~n

excitation coefficients. Examples are shown for the Benelux and for Great Britain-Ireland. The power to be delivered to a contoured beam antenna is compared with that to be delivered to a conventional parabolic reflector antenna to illuminate the same region. It is checked whether the calculated power distribution meets the WARe specification. The frequency dependence of the contoured beam radiation pattern has been investigated. Finally, attention is paid to the aperture distribution, the spillover, the beam efficiency and the gain of a contoured beam antenna.

Herben, M.H.A.J.

DESIGNING A CONTOURED BEAM ANTENNA.

Eindhoven University of Technology, Department of Electrical Engineering,

Eindhoven, The Netherlands. December 1979. TH-Report 79-E-104

Address of the author:

Ir. M.H.A.J. Herben,

Telecommunication Group,

Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 MB EINDHOVEN, The Netherlands

Acknowledgement

The author wishes to thank Dr. E.J. Maanders for the useful discussions with respect to the underlying subject and Mrs. L. de Jong-Vriens for typing the manuscript.

1. General introduction

High gain spacecraft antennas will in the future employ contoured beams, i.e. beams giving a sharply contoured footprint on earth. With such antennas it is possible to concentrate the majority of the transmitted power on board the satellite uniformly over a defined area and to minimize the power outside that area. Dion et al. [1] designed a variable coverage satellite antenna consisting of a lens and a feed cluster.

Duncan et al. [2] proposed, by using reflector antennas, to illuminate a defined region with a composite beam consisting of spot beams, each spot beam being combined with the adjacent spotbeam(s) at their -3dB contours. The design discussed in this report uses some of Duncan et al.'s ideas, but with narrower contours [3], [4].

Moreover, practical circumstances with respect to the launch vehicle to be used have been taken into consideration. This means that the maximum diameter of the reflector should be less than 2.5 metres if the Ariane rocket is used and 4.5 metres if Space Shuttle is used. The frequency applied is 20 GHz.

2. The coordinate transformation 2.1 !~~~~9~~~!~~

A circularly ~arabolic reflector antenna with only one feed will have a main beam with circularly symmetrical properties if the feed, too, has these properties. If we replace this main beam by a circular cone with the vertex at the satellite the cone pointing to earth, the area Al of earth and cone will generally not be circular (Fig. 2.1). On the other hand, the area A2 on a sphere that has its centre at the vertex of the cone, will be a circle. In order to simplify calculations regarding a certain region of the earth (e.g. the Benelux), a coordinate transfor-mation has to be carried out. This transfortransfor-mation enables us to

determine for a point P on earth with known longitude and latitude the

coordinates

e

an , of a spherical coordinate system around the satellite

(Fig. 2.2). In this way the contour of the region on earth is found transformed on a sphere with radius I and its centre 5 which represents the satellite. The corresponding region can be covered with (circular) spotbeams which touch each other at their -3 dB contours (Fig. 2.3). We suppose that the satellite antenna axis pOints to the "centre" 0 of the contour. The angle between this direction and the equatorial plane is 9

0, In the present chapter equations will be derived to fulfil the coordinate transformation.

2.2 !~~~~~~~~~!~~_~S~~~!~~~

Fig. 2.4 shows the latitude a-longitude S system of the earth with centre M and a 1, 91

, $' satellite coordinate system. The satellite is assumed

to be in a geostationary orbit above a point A on the equator. Point A is the reference point for our longitude 8. In other words, the longitude in this point is supposed to be O. The latitute a is the ordinary geographical latitude. For the signs of a and S see Fig. 2.4. First of all we have to determine the

e' ,

~I coordinates of a pOint P on earth by a given longitude and latitude.

The angle y is given by

cos

y

= cos a . cos

a

(2 • I )

PS2 ₌ Mp2 + _{MS2 - 2 MP MS} cos y 2 _{+ r,2} 2 ar' = a

-

cos y 2 a (1 + R2 - 2R cos y) ) 2 1 PS a\;R

-

2R cos Y + with:

a radius of the earth,

r' distance from the centre M of the earth to the origin S of the satellite coordinate system,

R = r'

a

The sine equation for triangle MSP results in sin

a'

MP

-~

PS

With eq. (2.1), (2.2) and (2.3) we obtain for sin

a'

, 2 2 sin

a'

=

-'\,,;1:;;:-=C:O:S==CI=C:O:9=:~====

\ ' 2 fR - 2R cos

CI

cos ~ + 1 (2.2) (2.3) (2.4 )

To calculate the angle ~' we introduce with M as origin. The x I and y I axes

a new coordinate system x " y I

a a

a a are tangential to the Xl and

y' axes respectively (Fig. 2.5).

MP

1 is the projection of MP on the plane V. P,P"" is the projection ,of PQ on the plane V and PQ is tangential to the plane V.

Hence

PIP,t •• = MP", = PQ = a sin Q. (2.5)

MP"" is the projection of QR on the plane V and QR is tangential to the plane V. Hence MP"" = QR a cos

CI

sin ~ Therefore MP" I I tan ~ , = :;;;:---a MP, I I sin

S

- - - = tan ~' tan

CI

(2.6) (2.7)

Suppose that the antenna points to a pOint 0 (a = a

O;

e

= a) on earth. We now define a new satellite coordinate system x, y, z (Fig.2.6) • The new coordinates axe given by

x = sin

a

cos $ = sin

a'

cos $' cos

ao -

cos

a'

sin

ao

y sin

a

sin $ = sin 8 ' sin $' (2.8 )

Z ::I cos

e

= cos 8' cos 8

0 + sin 8' cos $' sin 80

With the cosine equation on triangle MSP the distance satellite - earth SP is found to be

SP = r' COS 91

V

I 2 2 2

- a - r' (1 - cos 9')

with (using eq. 2.8)

cos 8' = [cos 8 sin 8 0 2.3 Calculations

---sin 8 cos cos 8 0 (2.9) (2.10)

The coordinate transformation described in this chapter has been worked out for the contours of the Benelux and that of Great Britain - Ireland. As the contours are relatively small, the opening angle at the satellite bounding the contours is very small as well (cos

a

~ 1); therefore, the contours may be mapped in one plane. Fig. 2.7 shows the transformed contour of the Benelux.

The satellite axis pOints to a point

a

on earth with geographical longitude 50 E.L.; 8

0 = 7.44 0_•

Fig. 2.8 represents the transformed contour of Great Britain - Ireland. In this case

geographical

the satellite axis is o

longitude 4.25 W.L.,

directed to a point 0 on earth with

a

8₀

=

7.65 • The following numerical values have been used:

a = 6378 km r'= 42162 km R = 6.61

3. Excitation coefficients of the feeds 3.1 Introduction

Let us now illuminate a region transformed to the e,~ coordinate system

by an antenna with more than one feed. If the excitation coefficients of all feeds are equal, the power distribution over the illuminated area

will in general not correspond to the requirements. This is caused by

interference between the spotbeams. Therefore, we have to modify the excitation coefficients in such a way that the resulting power distri-bution approximates the desired power distridistri-bution as well as possible.

We shall introduce optimization points, i.e. the points for which the

desired power is defined. In this chapter a method of calculating the excitation coefficients will be developed.

3.2 ~~~~~~~~!~~_~~_~~~_~~~!~~~!~~_~~~~~!£!~~~~

Suppose we have n feeds and m optimization points.

th Define the far field in point SP

m, em' $m caused by the n feed of the

reflector antenna as

ER (SP , Bm' $ _{n m m} 1 + jEI (SP , B , $ _{n m m m} 1

with

ER (SP ,

n m B m $m

,

1 the real part of E (SP , B n m m

,

$m 1 EI (SP , _{em' $m l the imaginary part of E (SP , em' $ml}

n m n m

The total far field in the point SP , Bm' ~m is

with E(SP , m Bm' $ml ER(SPm, Bm' ~ml EI (SP m' 8m, ~ml = m ER(SP , m em' ~ml + j EI (SP , m em' $ml

the real part of E(SP , 8 , $ 1

m m m

the imaginary part of E(SP , 8 ,~ 1 m m m th

The complex excitation coefficient of the n feed is

COEF

=

COEFR + j COEFI

n n n

with

(3. 11

(3.21

COEFR n COEFI

n

the real part of COEF n the imaginary part of COEF

n

We shall also define matrices ReE and ImE as

r'

,~,.:".

~1)

...

ER n(5P1,.91,

.,']

[ReEJ = ER1 (5P , 9 , _~m)

...

ER (5P , 9 _{m' ~m)} m m n m =

[EI1(5P1'~91' ~1)

.•••....••..

EIn(SP1'~91' ~1)]

EI1 (SP ,

a ,

~

) ...

EI (SP ,

e ,

~

)

m m m n m m m (3.4)

According to the complex arithmetic, the relation between Eq. 3.1, 3.2, 3.3 and 3.4 is given by [ReE] [-ImE]

[T']

["'"''

:'"

"

']

COEFR ER(SP ,

a ,

_~m) n m m (3.5)

H

[ReE]

[OO~"']

COEFI n

["

''',.

:'"

EI(SI' ,

e ,

m m

"

]

~m)

The desired total far field in an optimization point 5P

m,

am' ¢m

is EO(SP ,

a

m' ~m) = EOR(SP ,

e

m' ~m) + jEOI(SP ,

a

,

~m) (3.6)

m m m m

with

EOR (SP ,

e

,

_~m) the real part of EO(SP ,

e

_{m' ~m)}

m m m

EOI (5P , 9

,

_~m ) the imaginary part of EO(SP ,

e

,

_~m)

m m m m

[ReE] [-rmE]

[~"']

[roW" , ' :"' '

_','

]

COEFR EOR(SP ,

e ,

$ ) n m m m = 0 [rmE]

[~,

]

[T']

[ID"""

j'"

"']

(3.7) COEFr _{EOr(Sp , 8 , $m)} n m m With +jkSP E '(SP , 8 , _$m) E (SP , 8 , $ ) . e m n m m n m m m and (3.8) +jkSP EO' (SP , 8 , $ ) = EO(SP , 8 , $ ) • e m m m m m m m Equation (2.7) becomes

H

[-'~J

[T']

OOEFR

[roR' ''', ' :" '

_{EOR' (SP , 8 , $m)}

.,]

n m m = [rmE] [ReE' ]

[T']

r" "'" :""

,,]

(3.9) OOEFr _{EOr'(SP , 8 , $m)} n m m

Equation (3.9) will have one exact solution if the number of feeds n is equal to the number of optimization points m. Generally m > n.

In this case we have to determine the least squares or the minimax solution of Equation (3.9).

The least squares solution of Ax - b

=

0 with A a known m' x n' matrix,

b a known m' vector and

x

an unknown n' vector is given by

m' n' 0 min

L

i=l (b -i 2

J:

AijX J.) j=l (3.10)

The minimax solution is

min max i n'

L

j=l (3.11)

In the next chapters we shall make use of the least squares solution. The phase of EO(SP ,

a ,

$ ) is so chosen that the destructive interference

m m m

between individual spotbeams is minimum.

4. Calculation of the contoured beam antenna pattern

4.1 Introduction

First the radiation pattern of a defocused parabolic reflector antenna has to be determined to calculate the radiation pattern of a contoured beam antenna. In this chapter we shall use a scalar theory to calculate the radiation pattern. The symmetrical parabolic reflector antenna will be considered. Because of the blockage in such a system, the same w111 be done for an offset configuration. The important magnitudes required for the design e

3dB'BOF and (F!O)min are determined. After that we shall design contoured beam antennas for Great Britain-Ireland and the Benelux.

4.2

!~~_~~~~E!~~!_E~E~~~!!~_E~~!~~~~E_~~~~~~~

4.2.1

!~~_~~~~~~~~~_~~~~E~~~!_E~E~~!!~_E~~!~~~~E_~~~~~~~

Fig. 4.1 shows the geometry of a defocused symmetrical parabolic reflector antenna. According to Ruze [5] the scalar :far field E(R,e,~) is given by

with ·2 = u -jkR je

AR

0/2 'k (A')

J

f(r,~')eJ rA cos ~

-s

rdrd$' 2uu s

o

---M(r) cos ~ + u sin ~ 2 tan

B

= u 8 Z u cos ~ - u !M(r) s sin

e

u

=

8

IF

s x (8 2 + 8 2) I 2F (Petzval surface [5], [6]) x Y 8 Y

=

0 (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7)

M(r)

=

1 + (r/2F)2

In case f(r,$') = fer) and £x ~ £t cos a, Ey becomes with g(e,$) 2

=

u tan

S

D/2 2~

f

f(r) JO(krA)rdr

o

2uu s - - - cos M(r) ($ - Il) +

u sin (<I> - Il)

u s 2 2 M(r) u cos ($ - Il) - u /M(r) s u s Et / F 2 E t /2F 2 \ Ex + E _Y 2 E sin a equation (4.2) t

The approximations

£

« 1 and cos

e

~ 1 have been introduced. In our

p

calculations we shall use the aperture distribution function [7]

f(r) = q + (1 - q) (1 - (2r/o)2)p (4.8) (4.9) (4.10) (4.11 ) (4.12 ) (4. 13) (4.14 )

Fig. 4.2 shows the computed radiation patterns of a symmetrical parabolic reflector antenna (f(r)

=

1, 0/,

=

510, F/O

=

0.67, $

feed displacements along the negative x axis given by EX

(k: number of beamwidths scanned).

0) with several k

*

0.75 ,

In this figure the power is normed to the power

Po'

radiated in the

e

=

0

direction by the same antenna with a focused feed. From the figure we see that

The left sidelobes become higher (coma),

The right sidelobes become lower,

The first right sidelobe disappears completely for a given feed displacement (k = 6),

The 3 dB beamwidth is almost constant, The maximum gain decreases little.

4.2.2 !~~_~_~~_e~~~~!~~~

We have seen in section 4.2.1, that the field strength Ig(8,$) I is given by

D/2

J

f(r) Jo(krA)rdrl

o

Under focused condition this becomes D/2

Ig(8) I

=

12~

J

f(r) Jo(kr sin 8)rdrl

o

1

=

12 w(_D2)2 "

J

f{r') JO{r'

T

~D s~n 8)r'dr' .

I

o

with r'

=

2r/D. To calculate 8 3dB we have to solve

For small angle 8

3dB the general solution becomes

J-sin 8

3dB

=

83dB

=

const

D

Fig. 4.3 shows canst as a function of P, q being a parameter.

4.2.3 ~~~_~~~!~~!~~_!~~~~~

The beam deviation factor BOF is defined as

[5]

with sin 8 BDF = _ _ -,m~ E t F The angle 8

m is the 8 belonging to Ig(8) I max •

(4.15) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21)

E

t

For small feed displacements ( « 1 ) the BDF is given by [5] p with BDF 1 3

f

f(r')r'

o

M(r') dr'

o

1

f

_fer'

_)r' 3d ' _r M(r')

=

1 +

(r~~)2

Fig. 4.4 shows BDF as a function of F/D, p and q being parameters. For a large F/D (plane mirror) the BDF approximates 1.

(4.22 )

(4.23)

4.2.4 ~~!~~!~~_e~~~~~~_~~~_~~~~~~~_~LQ~_~~~_~!~~~~~~_~~_~_~~~~_~~~_~~~_~e~~~~~~

distribution

As we have seen in Chapter 1, the feeds of a contoured beam antenna are so arranged that the -3 dB contours of adjacent feeds touch each other in the far field.

We now define (F/D) . as the F/D of a parabolic reflector in which the

m~n

feeds touch each other as do the accompanying -3 dB contours in the far field. For this (FlO) i we find

m n

d BDF

(F/D)min

=

I

const (4.24)

where d is the diameter of the feed, which is mostly between 0.75 A and

1.5 A [2].

In Fig. 4.5 (F/D) . has been represented as a function of d/A, p and q m~n

being parameters.

4.2.5 ~~~_~~~~~~~~9_e~~~_~~~~~~~_e~~~~~~

Fig. 4.6a shows the radiation pattern of a symmetrical parabolic reflector

antenna with 19 equally excited feeds arranged hexagonally and located on the

Petzval surface. It appears from this figure that the power distribution over the illuminated part of the sphere around the satellite is not very uniform (-6.5 dB in the

e

=

0 direction).

Fig. 4.6.b illustrates the radiation pattern again but now with improved excitation coefficients. The ripple is lower than 0.5 dB. From these two figures we see that it is very useful to optimize the excitation

coefficients.

Fig. 4.7.a and 4.8.a show the normalized power distribution over Great Britain-Ireland using a symmetrical parabolic reflector antenna with

reflector diameter D

=

2.5 m and D

=

4.5 m. The total number of feeds is 7 and 14, the optimization points (see also Fiqs. 4.7.b and 4.8.b) are 15 and 29 respectively.

An attempt to design also a contoured beam antenna for the Benelux failed

because of the D = 4.5 metres limit. However, as soon as it is possible

to launch larger reflector diameters, it will also be pssible to illuminate

the Benelux with an appropriate contoured beam antenna. In the meantime we

have designed a contoured beam antenna with a lO-metre diameter reflector

for the Benelux.

In Fig. 4.9.a the normalized power distribution over the Benelux has been

represented using a symmetrical parabolic reflector antenna with reflector diameter D

=

10 m. illuminated by 18 feeds.

The -3 dB contours of the individual spotbeams and the 38 optimization points are mapped in Fig. 4.9.b.

All computed powers are normed to the power.PO(dB). This power is related to the power radiated in the a

=

0 direction by a similar antenna with one feed in the focus, the excitation coefficient being equal to 1.

4.3.1

!~~_~~!~~~~~~_~!~~~~_E~E~e~!~£_E~~!~£~~E_~~~~~~~

The scalar far field E(R,a,~) of a defocused offset fed parabolic reflector antenna with the geometry as shown in Fig. 4.10 illuminated by a feed with illumination function Gf'(W,,) is given by [8].

E(R,a,~) jWIJ -jkR

_[(

E )" P TJ"

e-2jkFg(a,~)

= - - e 21TR IJ 21T with 21T _W [G_f '

(W,~)l"

-jk(p·-p.a_R-2F) a 2 g(a,~)

J

J

e p sin WdWd~ 0 0 p'

One has to determine p.a

R, p' and Gf ' (W,,) to calculate the far field.

The scalar product

PoaR

is found to be

p.a

R = p(sin W cos, cos Wo + cos W sin

W

O) sin a cos ~

-p sin W sin , sin a sin ~

-p (cos

w

cos Wo - sin W cos , sin

W

O) cos a .

(4.25)

(4.26)

The distance pI from feed to reflector is

p'

=

{[p(sin

~

cos

~

cos

~o

+ cos

~

sin

~o)

+ £x12 +

2 [£ - P sin ~ sin ~l +

y

"

[£z - p(cos

~

cos

~o

-

sin

~

cos

~

sin

~0)12}

£

Using the approximation « 1,!q. (4.28) yields

p

p'

=

p + £x(sin ~ cos ~ cos ~O + cos ~ sin ~O)

- £ sin ~ sin ~

y

(4.28)

(4.29)

The displacement vector e: can also be defined in the XII, y", z" CDordinate

system

£

=

The relation between e: II' e: II' e: " and £ Ie:, £. is

x y z x y z

E

= -

E "

Y Y

£

Z EX" sin

*0 -

E ZII cos

tPO

Substitution of (4.31) in (4.29) yields

pi = P + e: " sin 1JJ cos ~ + e: " sin 1JJ sin F,: + E

z" cos W

x y

(4.30)

(4.31)

(4.32)

The exponential power in the integrand (Eq. 4.26) now becomes (without the constant factor -jk)

p' - p.a

R - 2 F p + e: X II sin

W

cos ~ + e: Y II sin

W

sin ~ + Ezlt cos

W

-p(sin ~ cos ~ cos ~O + cos ~ sin ~O) sin

e

cos $

+ P sin ~ sin ~ sin

e

sin $

+ P(cos ~ cos ~o

-

sin ~ cos ~ sin ~O) cos

e

Introducing the approximation cos

e

= 1 and using the relation

p

=

2F/(1 + cos ~O cos ~ - sin ~O sin ~ cos~), Eq. (4.33) may be written as

p' - p.a

R - 2F EX" sin ~ cos ~ + £ "

Y sin ~ sin ~ + EZII cos ~

- p(sln ~ cos ~ cos ~O + cos ~ sin ~O) sin e cos $

+ p sin ~ sin ~ sin e sin $ (4.34 )

To calculate _{Gf '} (~,~), we define a new shifted coordinate system ~', ~' (see Fig. 4.11).

In this system the cosine illumination function of the feed is given by

The relation between

W,

p' sin ~' cos ~' P' sin ~' sin ~' P , _{cos tjJ'} n 2(n+1) cos ~' = 0 ~ and ~'

,

~' coordinates p sin _~ cos ~ + E x" p sin tjJ sin _~ + £ y" p cos tjJ + £ z"

Eg. 4.35 and Eg. 4.36 yield

= 2 (n+ 1) { 1 p' n (p cos ~ + £ ,,)} z ~'

>

is 7T 2 7T 2 (4.35) (4.36 ) (4.37)

Fig. 4.12 shows the normalized power and phase of g(e,O) and gee,

I)

with

different feed displacements along the Xl, x" and y" axis.

To get an impression of the total radiation pattern some radiation patterns

are mapped for all values of ~ (see Fig. 4.13).

In this figure the power is narmed to the power PO' radiated in the 8 = 0

direction by the same antenna with a focused feed. The decrease in the

maximum gain caused by defocusing seems to depend on ~O and

Wa"

Therefore, 1

"

20 log (Ig(e,o) I) - P , and 20 log ( gee, -2') I) - PO' have been

max 0 max

calculated as functions of k for several tjJO' tjJa combinations (Fig. 4.14).

For feed displacements in the x", y" plane (offset focal plane [9]) BDF is a constant. The BD}<~ belonging to a feed displacement along the x' axis is smaller than the BDF belonging to feed displacements along the y' (y") axis.

The 3 dB beamwidth is almost constant for feed displacements along the x" and y" axis.

The radiation patterns for feed displacements along the y" axis are in

good agre~ment with the radiation patterns of a defocused symmetrical

parabolic reflector antenna.

For feed displacements along the x" axis we see a rather fast degradation of the radiation pattern. We have no deep zeros any longer, the main lobe gets a 'shoulder'.

The decrease in the maximum of 20 log (lg(6,0)

I) -

PO' is almost equal to the decrease in the maximum of 20 log (lg(6,

T)

I) -

PO'.

The larger the offset angle, the faster the decrease in the maximum by

a given lJi • a

With regard to the main beam we notice that under the focused condition there exists a phase plane, which comprises the y axis and which is at an angle to the x axis, depending on ~O (in case lJi

O

a

this angle is 0) [10]. If we displace the feed along the x' axis the phase plane will shift but the angle to the x axis remains the same. However, i f we displace the feed along the x" axis the phase plane will almost coincide with the phase plane under the focused condition.

4.3.2 Power n of the cosine illumination function

---Pagones [8] has derived an expression for the gain factor g of an offset parabolic reflector antenna irradiated by a feed with a cosine illumination function. He found

g 2 (n+l)

(_C_O_S_lJi-"~'-i-:-:-:-S-ljJ.:!.a

r

C/

a

cos

~

lJi (os

(4.38) sin

In the following we shall choose the pO>ler n at a given ljJo and ljJa in such a way that g is maximum under the focused condition. Fig. 4.15 shows gmax as a function of ljJa' ljJo being a parameter. For the n belonging to gmax as a function of ljJa see Fig. 4.16.

4.3.3 The 3 dB beamwidth

---From Eq. 4.26 and p ~ 0 we obtain 2n ljJ"

J J

o

0

'ITO

exp (j A sin 8 function (ljJ, ~, ~, ljJO' ljJa))

p 2 p sin ljJ dljJd~ To calculate 8 3 dB' we have to solve

I

_g(

_""""f

no _{sin (., 8}

I I

I

-0.15 3dB) ,~) - g(O) . 10 = 0 The solution appears to be almost independent of ~.

For a small angle 03dB the general solution becomes A

sin 8

3dB

=

const 0

Fig. 4.17 shows const as a function of ljJa' ljJo being a parameter.

4.3.4 Beam deviation factor

The BOF of an offset parabolic reflector antenna is given by [11].

BOF

The Xli, y" plane is the offset focal plane.

(4.39)

(4.40)

(4.41)

(4.42)

Fig.4.18 shows BOF as a function of W

a' Wo being a parameter. It has been found in lit. [9] that for a given set of ljJa and ljJO the ratio of the BOF to the

Flo

is constant for a given ~ and a given edge illumination. This

a

means that the BDF of an offset reflector can be calculated from that of a front fed reflector with the same subtended angle using the following equation

thus (BDF) offset (F/D)offset (BDF) s~ (F/D) s~ (BDF) s~ cos ~ a ~ (BDF)offset COS ~a + (4.43) + 1 cos ~o (4.44)

We calculated (BDF)S~ from (BDF)offset as a function of ~ .

using Eq. (4.44). Fig. 4.19 shows this (BDF)

s~ a

4.3.5 ~!~~!~~_e~~~~=~_~~~_~~~~~~_~~e~=~~~~_~~S~=_~a max ~~2_~~~_2~~~~~~E_~_~~

a feed.

According to Eq. 4.24 the relation between (F/D) i and the diameter of a m n feed d is given by d

A

const (F/D) . I BDF m~n (4.45)

In Section 4.3.4 we have seen that the ratio of BDF and F/D is independent of ~O at a given ~a. From Fig. 4.17 i t is seen that const is almost

independent of ~O. The result is that dlA is only a function of ~a. Fig. 4.20 shows ~ as a function of d/A.

a max

4.3.6 ~~~~~_9!_!~~_~~~!~~~_!9!~!_!~~_~!~!~_~2~J§Pm'

8m,

~m)

In Section 4.3.1 we have seen that with regard to the main beam there exists a phase plane parallel to the y axis, and which is at an anqle to the x axis depending on ~O (in case ~O ~ 0 this angle is 0). In the case of 2 feeds

(Fig. 4.21) we have to choose the phase of EO' (SP , 8 , ~ ) in such a way

m m m

that in point SP

1, 81, $1 the phase of E)' (SP1, 8), $) is equal to the phase of E

2' (SPt, 8t, ~1). In other words, the phase planes of spotbeams and 2 should coincide. Otherwise destructive interference will occur.

4.3.7 !~~_S9~!9~E~~_~~~~_~~!~~~~_E~!!~E~

Fig. 4.22a shows the normalized power. distribution over Great Britain -Ireland, using an offset parabolic reflector antenna with reflector diameter

o

= 4.5 m. The number of feeds and optimization points are 14 and 29 res-pectively (see also Fig. 4.22b). We notice that this power distribution is almost equal to the power distribution found in section 4.2.5 (Fig. 4.8).

5. Further investigation of some properties of the contoured beam antenna

5.1 Introduction

---In this chapter the power delivered to the contoured beam antenna is compared with that delivered to a conventional antenna to obtain the same power density in the illuminated region in both cases. To check whether the calculated power distribution meets the WARe specifications we shall calculate the sidelobes of the contoured beam antenna pattern. After that the frequency dependence of the radiation pattern will be studied.

Finally, attention will be paid to the aperture distribution, the spill-over, the beam efficiency and the gain of a contoured beam antenna [12].

5.2 £~~E~!!~~~_~i_£~~~~~!~~_e~~_~~~~~~~~_~!~h_£~~~~~~!~~~!_~~i!~£~~!_~~~~~~~~ The overall gain G(O,O) of a reflector antenna, irradiated by a feed with a cosine illumination function is given by

G(O,O) g

(rr~y

(5. I) with Eq. 4.38 G (0,0) 2 cos tjJ

(:!L£)

2 (n+l) ( 0 A sin n/2 ( sin tjJ ) 2 cos tjJ -co-s-tjJ-o-+-c-o-s-tjJ . dtjJ) (5.2) ,I, th

We notice that g is determined completely by tjJO' o/a and n. Let the n feed of a contoured beam antenna radiate a power PIn and let the diameter of the aperture be 0

1° We now illuminate the region by a conventional reflector antenna with one feed in the focus. The wavelength

A,

tjJO' tjJa and the radiation pattern of the feed are equal to those of the contoured beam antenna. The reflector diameter D2 of the conventional antenna should be such that the -3 dB contour of the main beam comprises the entire region.

Let further the feed of the conventional antenna radiate a power P2. We

use the same normalization as we used for the contoured beam antenna. The ratio P to P

2 is now In

The ratio between the total power P

1 supplied by all n feeds to the reflector and P2 now is

PI

(:~ y~

I

COEF nl2 (5.4)

P 2

Some results are found in the table below

Figure _{D1 D2} PI P 2 4.7 2.5 0.77 0.24 4.8 4.5 0.77 0.18 4.9 10 1.67 0.35 4.22 4.5 0.78 0.15 5.3 !~~_~~~~_~2~~!!!~~~!~~~

To check whether the calculated power distribution meets the WARC

specifi-cations (see Appendix A), the sidelobes of the contoured beam antenna pattern have to be known. Fig. 5.1 shows the main beam and the sidelobes

of the contoured beam antenna radiation pattern for Great Britain - Ireland

(D

=

4.5 m) described in Section 4.2.5.

In fact, the WARe specifications are drawn up for antennas with circular and elliptical beams. When applying these specifications to the radiation

pattern represented in Fig. 5.1, too strong requirements are imposed on

the directions 2850 < ~ < 310°.

In these directions 63dB/2 is very small, (see Fig. 5.2). To get over this problem, the WARC specifications can be applied to an ellipse enclosing

the -3 dB contour of the contoured beam antenna pattern. Fig. 5.3 shows that our power distribution meets these specifications. According to international norms every country has its own ellipse. Fig. 5.4 shows the combined ellipses for Great Britain and Ireland. We notice that our power distribution also meets the WARe specifications applied to these ellipses.

5.4 ~~~S~~~~~_~~e~~~~~~~_~!_~~~_E~~!~~~~~_2~~~~E~

In Figure 5.1 the normalized power distribution over Great Britain -Ireland is shown using a symmetrical parabolic reflector with 14 feeds

(D

=

4.5 m). The frequency applied is 20 GHz. Generally, one will not

transmit at only one frequency : a frequency band is used. That is the reason why it is very important to know the frequency dependence of the radiation pattern. Figs. 5.5 and 5.6 contain the normalized power diS-tributions at 18 and 22 GHz. All computed powers are normed to the power

PO(dB). The latter is related to the power radiated in the

e

=

a

direction by the same antenna with one focused feed at 20 GHz the excitation

coefficient being equal to 1.

We notice that in the case of such a great relative frequency change (10%) the deformation of the original power pattern is small.

5.5 ~e~~~~~~_~!~~~!e~~!~~

The scalar far field of a defocused parabolic reflector antenna is given by

(Eq. 4.35, 4.26) E(R,8,cp) = with jw~ 27TR -jkR e e -2jkF g(e,~) -jk(p,-p.a R-2F) e P 2 sin .pd.pdl;.

If we use a contoured beam antenna, this formula becomes

(5.5) (5.6)

f

COEF i e -jk(P_i '-2F)

I

jkp.a_R e p2sin ¢d¢dl;

o

The plane z

=

a

is the aperture plane. The approximations £ < < 1 and

P

cos

e

~ 1 are now introduced. For the aperture distribution we find

I

COEF i [G fj ' (¢,I;)]" -jk

P

·;i/P

I

F(.p,I;) =

I.

e i _Pi with G

f .' (.p,I;) = 2 (n+ 1)

I

p1i' ( P cos .p+E"·)r

~ z ,~

(5.7)

(5.8)

(5.9)

In section 4.3.7 we calculated the power distribution over Great Britain -Ireland using an offset parabolic reflector antenna with D

=

4.5 m.

The normalized power and phase of the aperture field of this antenna is shown in Fig. 5.7. We notice that if F(r,$') 0, the phase arg F(r,$') is not defined. The coordinates rand 4>1 are the polar coordinates in the aperture plane. The latter is no longer an equiphase plane. So i t will be

very difficult to produce this aperture distribution with one feed.

5.6.1

~e~!!~~~~_~!_~_~~!~~~~~~_~~~~e~!~~_~~!!~=~S~_~~~~~~~

For the geometry of a de focused reflector antenna see Section 4.3.1,

Pig. 4.10. The unit vector in the direction of the incident ray is a , and _p the distance from feed to reflector pl. The spillover may be written as

[12] • spillover

=

1 - 4;

f

o

G f ' (1/1,1;) is given by Eq. 4.37. a n ap ' 2 p sin ljIdl/ldl; a .a n p

The calculation of a .a I and a .a

p will be performed in the X, y, z

n P n

(5.1e)

coordinate system. The equation of the paraboidal surface in these coordinates is

g(x, y, z)

=

x 2 + y 2 - 4F(z + F)

=

0 (5. 11)

The unit vector normal to the reflector is

[

]

V

x ₁ a = -2.L = p n l'1gl

_~P

_{\ ' Px} / 2 _{+ py} 2 + 4p2 (5.12)

The ratio of the two scalar products becomes

a .a _p' p

a

.Pl

P a .p + a £

~

(1

'1g.;:

).

n n n n = - - - - = = + p'

- -

p' P' "g.P a .a an .p a .p n P n (5.13)

With Eq. 5.10 the spillover yields

21T I/Ia

j

1

f

2 (n+1)

-o

P' spillover = 1 -

f

41T 0

"g.£)

+ - - _ sin "g.P (5.14 )

The spillover has been calculated for feed displacements along the x" and

beam is scanned k beamwidths. These feed displacements are given by

F

E

=

k const - A/BDF

D (5.15)

Fig. 5.8 shows the spillover of a defocused symmetrical parabolic reflector antenna as a function of k for feed displacements in the XII y" plane ,

D = 4.5 m (Space Shuttle) and A = 1.5 cm.

Fig. 5.9 shows the same but now for D 2.5 m (Ariane rocket) and A

=

1.5 cm. Fig. 5.10 represents the spillover of a defocused offset parabolic reflector antenna as a function of k for feed displacements along the x" and y" axis, D

=

4.5 m and A

=

1.5 cm and Fig. 5.11 for D

=

2.5 m

We can also determine the spillover as a function of

combination.

and A = 1.5 cm.

E

D

by a given ~O' ~a Fig. 5.12 illustrates the spillover of a defocused symmetrical parabolic reflector antenna as a function of Et/D,and Fig. 5.13 that of a defocused offset parabolic reflector antenna as a function of £x,,/D and Ey"/O.

5.6.2

~E!!~~~~~_~i_~_~~~~~~E~~_e~~~_~~~~~~~

It is easy to see that the spillover of a contoured beam antenna is given by

2:

1 COEF 12 spillover spillover c.b.a. = n n n

2:

ICOEF 12 (5.16) with Spillover c.b.a. Spillover n COEF n n n Spillover of the Spillover of the

contoured beam antenna,

th n feed,

th

The complex excitation coefficient of the n feed.

Using Eq. 5.16, the spillover of the offset contoured beam antenna for Great Britain and Ireland 1s 8.132%.

5.7

~~~_~ii!~!~~~~

We define the beam efficiency as the ratio of the utilized power to the total power radiated. The utilized power equals the solid angle bounding the region to be illuminated multiplied by the minimum power density within that solid angle. Usually this minimum power density is half the maximum power density ,(-3dB requirementsl.For the '::learn efficiency of a reflector antenna we find

with P m P T ." P m

o

f

21T

o

f

e

(~) c sin ed8d~

the maximum power density in the the power radiated by the feed,

(5.17)

area,

e (~) the angle e c(~) at which, at a given ~. the contour of the illuminated c

region is seen by the satellite (Fig.

The relation between Pm and P

T is given by

P

m 41T PT g

(~0)2

A

Equations 5.17 and 5.18 yield

21T e (~)

~(1TO)2

f {

81T A 0 0

The maximum beam efficiency

( 1TO)2 2 DB =

~

- sin .. max A sin eded~ with 8

3dB the half-power beamwidth. For small

e

3dB Eq. 5.20 becomes DB

=

g max 2 1T 32 2 canst 5.14). (5.18) (5.19) (5.20) (5.21 )

The maximum beam efficiency is completely determined by ~O and ~a' Fig. 5.15

shows 11

Bmax as a function of

W

a,

Wo

being a parameter. It appears that

n = 33% i f we use a feed with a cosine illumination function. The beam

Smax

efficiency of the conventional antenna (0

=

0.78 m) to illuminate Great

Britain and Ireland is 8.4%. This is considerably less than n . We can

Bmax

also use Eq. 5.17 to compute the beam efficiency of a contoured beam

antenna. Pm is then the desired power distribution within the contour. P T is the total power radiated by the feeds. The beam efficiency of the contoured beam antenna for Great Britain and Ireland described in Section 4.3.7 is DB " 55%. This is larger than nand D of the comparable

B Smax

5.8 Gain

For the gain of a contoured beam antenna we find the expression

G(e,~) = 4Tr p(e,~) / PI (5.22)

with PI the power radiated by the feeds of the antenna.

We have normed p(e,~) to the power PO(O,O) of a similar antenna with one focused feed, excitation coefficient being I. The power radiated by the feed of this antenna is P

2 and its maximum gain is given by

(5.23)

g being the gain factor. The ratio of the power P

2 radiated by the feed of this antenna and the pcwer PI radiated by the feeds of the contoured beam antenna is

(5.24)

With Eq. 5.22, 5.23 and 5.24 the gain of the contoured beam antenna is found to be p(e,~) 4Tr PO(O,O) P₂ G(e,~) = PO(O,O) P 2 PI p(e,~) g C ; r I = (5.25) Po(O,O)

I

IcoEF 12 n n

As the distances from the satellite to the pcints inside the contour to be illuminated vary only little it is possible to obtain a rather accurate im-pression of the gain

To this we must add

10 log

I

gC D

/

from the normalized power distribution.

I

to find the gain G(e,~). This yields, for example in Fig. 4.22,

(5.26)

We could have illuminated Great Britain and Ireland by a conventional antenna with one focused feed. At 20 GHz an aperture diameter D = 0.78 m would have

been sufficient. The gain G(O,O) belonging to this antenna is 43.4 dB. We notice that che gain of the contoured beam antenna is larger than that of the

elm,

lrable conventional antenna.

6. Conclusion

Well-designed contoured beam antennas using reflectors offer the possibility of distributing large parts of the transmitted power uniformly over arbitrary regions.

The gain and the beam efficiency of the contoured beam antenna are

considerably larger than those of a conventional antenna. Therefore, the power to be delivered to the contoured beam antenna is much smaller than that to be delivered to the conventional antenna to obtain the same power density in the illuminated region.

Our calculated radiation pattern of a contoured beam antenna for Great Britain - Ireland (D

=

4.5 m) meets the 'modified' WARC specifications. For a relatively large frequency change (10%) the deformation of this radiation pattern seems to be small. Because of the relatively small feed displacements (large D/A) the spillover of the contoured beam antenna is not much larger than that of the conventional focused antenna.

The cross polarisation properties of the contoured beam antenna and the design of the array feed have to be investigated before deciding to employ this antenna.

References

1. Dion, A.R., Ricardi, L.J.,

"A variable coverage satellite antenna system",

Proceedings IEEE, Vol. 59, pp. 252 - 262, February 1971.

2. Duncan, J.W., Hamada, S.J., Ingerson, P.G.,

"OUal polarisation multiple-beam antenna for frequency re-use satellites", AlAA/CASI 6th Communications Satellite Systems Conference, Montreal,

Canada, April 5 - 8, 1976,

Published in: SATELLITE COMMUNICATIONS; Advanced Technologies, Ed. by D. Jarett.

New York: American Institute of Aeronautics and Astronautics, 1977.

Progress in astronautics and aeronautics, vol. 55, pp. 223 - 243.

3. Herben, M.H.A.J, Maanders, E.J.,

IISome Aspects of Contoured Beam Antennas",

lEE International Conference on Antennas and Propagation, pp. 184 - 188,

London, 28 - 30 November 1978.

4. Herben, M.H.A.J.,

"Het antwerp van een contoured beam antenna",

M.Sc.Thesis (Dutch), August 1978, Eindhoven University of Technology,

Dept. of Electrical Eng., Eindhoven, Netherlands.

5. Ruze, J.,

"Lateral feed displacement in a paraboloid",

IEEE Transactions on Antennas and Propagation, Vol. AP-13, pp. 660 - 665,

September 1965.

6. Rusch, W.V.T., Ludwig, A.C.,

"Determination of the maximum scan gain contours of a beam scanning paraboloid and their relation to the Petzval surface",

IEEE Transactions on Antennas and Propagation, Vol. AP21, pp. 141

-147, March 1973.

7. Sciambi, A.F.,

liThe effect of the aperture illumination on the circular aperture antenna. characteristics",

8. Pagones, M.J.,

"Gain factor of an offset fed paraboloidal reflector",

IEEE Trans. Antennas and Propagation, Vol. AP-16, pp. 536 - 541,

September 1968.

9. Ingerson, P.J., Wong, W.C.,

"Focal region characteristics of offset fed reflectors",

IEEE/AP-S International Symposium, pp. 121 - 123, Atlanta, June 10 - 12, 1974.

10. Herben, M.H.A.J., Maanders, E.J.,

"The radiation pattern in amplitude and phase of an offset fed paraboloidal

reflector antenna",

Archiv fur Elektronik und Obertragungstechnik, vol. 33, pp. 413 - 414, Oktober 1979.

11. Rudge, A.W., Adiatia, N., Jacobsen, J,

"Study of the performance and limitations of multiple beam antennas", ESTEC contract no. 2277/74 HP, final report, September 1975.

12. Herben, M.H.A.J.,

Intermediate report (Dutch), May 1979,

Eindhoven University of Technology, Dept. of Electrical Eng., Eindhoven, Netherlands.

APPENDIX A

Fig. A.I.! shows the reference pattern for co-polar and cross-polar compo-nents according to the WARe specifications for a satellite transmitting

antenna.

Curve A : co-polar component.

-12(% )2

3dB -30

Curve B : cross-polar component.

10 - (40 + 40 log -33 10 -(40 + 40 log for 0 , 8 , 1.58 8 3dB for 1.58 8 3dB < 8 ~ 3.16 83dB for 3.16 8 3dB < 8 for 0 , 8 , 0.33 8 3dB for 0.33 8 3dB < 8 , 1.67 83dB for 1.67 8 3dB < 8

FIGURES CHAPTER 2.

Fig. 2.1

Fig. 2.2

Noncircular area At on earth, circular area A2 on a sphere

around the satellite S.

The coordinates

e,

$ of a spheri.cal coordinate system around the satellite.

Figa 2.3 The transformed contour C'

\

\

Fig. 2.4

--- The -3 dB contour of a spotbeam •

. 4-...

I"\~

1.

'PLane

V

e

M ct~

l:'

\ \

,

~'

The latitude a - longitude

a

system of the earth and a

1, 8', ~' satellite coordinate system.

S

~I

Xl

Fig. 2.5: The xa'.

,

)(

...

Y I coordinate system.

a

Fig. 2.6: The new X, y, z satellite coordinate system.

)('

Fig. 2.7 Fig. 2.B

...

o ~ .•.

-'-

....

-.,

,.

,.

./ "

-

...

\...~.-. ~"'" ... '! • _{" ,}

'-

.

....

.I..

.,' " _.110 2'10 •

...

," .. 'ljIQ

.,

....

Ii<> .

,

....

The transformed contour of the Benelux.

-

.10

...

. 40

/.-.~:;. r ,"-'-',

\

. <....

.10 f·..-_· ...

·Z";".·.,

' ....

'J' -. . . . . ;-'. . . .... ' ..

>. . '

II . , . . ... .. ... L

a

'\.~-4 It» +.... • ... tnO ~ < .•. -

_...

-....

.••• _.,.J ":'. ... . . OIIII>j"'"

...

...

~

•.

".:::::~~.

_.-'-'

..

'-' _._/ 2'<0 • ~I ..

FIGURES CHAPTER 4. Fig. 4.1 -IQ Fig. 4.2 l(

,.

\ \

~'

~,

1>1

,.

,-,.

Ii

The geometry of a defocused symmetrical parabolic reflector antenna.

-.1.

120~(I~fl,O)lt -"P.~

_{. .}

_.

_.

_.

_.

_.

'''\0 ·-010

·-so

.-be.

The radiation pattern of a defocused symmetrical parabolic

reflector antenna, £ k * 0.75 A (k : number of beamwidths

"

_&

-b

-'< -7. 0

t

2~~.( \()(Bp~) ~ 'P~~ I~

".10-3

__ 9C'\hd)

·-20 ·-50

-.b .

-~ -~ 0 ~

20~:\~(~/O;~)~~:

.

'~*.1()-3

_e<.Ra.d.)

. -'10 ·-YO ·-60 -'I

_-11-

c;> _~

'1

20~ (\~ (~D)l)

-

~'

8

0 1'1 14 ~

.

.

.

.

.

. .

,~~.

Ie?

---.. eCIlad)

n

·-20 • -:loa

(\

1<=<>

·-YO

·-so

·-be

Fig. 4.2 (con'oinued)

-~

• 0 •

~

.

't .

~

. J

~~.(~~l~~~)~~i

.

'9

~\o-~

'-\0

-e(~d)

lfl

. -:\0

k~4

'-20 . -'10 '-50 .-~ ·-~o

l

'-30

k",s:

'-40 '-50

'-1.0

t

~O

L ....

{lq (Sp)l ') _

'P"

I \0 'll. 1'iiJ

%

\8 ;'0 2h 10-3

.

' .

. . . .

• -10

--9Ucc:/.)

'-50

I

Fig. 4.2 (cOntinued)

j

j

J

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

j

I

i

CO'flst

2./:'·

2.4' 2.2· 2.0·

1'& .

\.1,.

1.4' 1.2. I . o

t

2o\m,

(1Q.(ep)I)-?'"

-3

14 .

iJij.

I~

. 2.0 . 2.2. 2'-1 if 10

_GCRDA)

'-20 .-~ '-'10 ·-50

.-bo

Fig. 4.2 (continued) 2> Lj 5

8

-I =10 _0.5

=

10 10

--"1'

t

13DF

1.0'

0.9'

0.&·

o.l

o.b.

0.5 .

o.y·

o.?> .

o.~. 0.1· 0.11.

t'B'Df

1.0'

000'

0.&·

o.l

e.b·

0.1," 0.3· 0.1 •

CJ.==o

0:&

0:8

.

I:b

1.8 ~

F/D

0.4 1.0 1.2- 1.'1 2.0

.

I.e:. 1.2 1.4

i

(f/D) .

"

..

"

\.8·

I.b.

\. '1 . 1.2.. 1.0 •

0.8.

a.b.

c .... ·

0.2. O.Oi

o:b

0.8

I.~ .

\.b.

1.2. • \.0·

0.8.

Clb.

0.'1 . 0.2·

jtflD )"""

0.'1

oJ,

~

1.0 1.2 1.11 \.0

1.2.

1.'1

lin

a

1."1,

I.B.

2.0 2.2 2.~

!l.b

2.8

3.0

.

~.2.

d./~

-d)

_1'&

.

2.0 2.2. ~.Lj ~:b

_ri8

_3.0 3.2.

.

Fig. 4.5 (Flo) as a function of

dlA ,

P and q being parameters. min

i

lf'D) .

_\""'1

\.8·

I.b·

1.4· \.~. 1.0·

0.<6.

o.~ Fig. 4.5

'300.

Fig. 4.6a _0.1> ~=\C

d.n.

~ \.0 1.4 3.0

3.Q.

(Flo) . as a function of d/A, p and q being parameters. m~n

o

.bo

.80

.

8tRD.d.)

6

i"O-~_ '\00 200

Radiation pattern of a symmetrical parabolic reflector antenna

with 19 equally excited feeds, arranged hexagonally.

--- - 3dB contour of one spotbeam

300.

180.

'l.bo·

140 .

Fig. 4.6b

3.00.

280.

20

~

-!lo

'32.0

₄₀

220

lLie

200

120

<p

(b)

be

.8e

b

'

lOIn

-~

_

G(god)

. Ice 120 -3dB contour of one spotbeam. $ optimization point. - - - lines of constant power.

Radiation pattern of a symmetrical parabolic reflector antenna with 19 feeds arranged hexagonally with improved excitation coefficients.

o

20

.bo

.&

~.-~

.(,,~ c·"---~-.~-:':'-:j '100

---~.-"

-

._,-,-240'

~-=-\~I)---~

.

120 Fig. 4.7a -20 ~2.0 . Ibo 140 _{transformed contour of}

Great Britain - Ireland . lines of constant power.

The normalized power distribution over Great Britain - Ireland

using a symmetrical parabolic reflector antenna,

300.

QSo. "

Qbo·

240 .

Fig. 4.7b 0

340

20

~

_~C»)

;;20.

-

_.

---... .~o ,

,

,

,

,

I

.bo

,

,

'-, "

,

y /

,

,

,

I

.80

,

I " I:_--~"

,

_{.' -3}

e(~M)

.

,

,CXI*\O

..

,

,

, I , _{. \00} ' i _/ r

,

'

' , _'IQo

,

/ _{-3dB contour of}

,

spotbeam. one

Qlo'

. 140

IIR optimization point.

- - - transformed contour

~OO

_ibo

of Great Britain

-\80

Ireland.

The normalized power distribution over Great Britain - Ireland

using a symmetrical parabolic reflector antenna,

D

=

2.5 fi, P

=

1, q

=

10- 0 . 5 , F/D

=

0.6, freq.

=

20 GHz.

o

,,40

10

'212.0

280.

. 240'

I"ig. 4.8a - 0

. 140

200

. ,be

,go

80

_~

8C"l.ad)

It It> _ ·100

• 120

_{transformed contour} of Great Britain -Ireland. lines of constant power .

The normalized power distribution over Great Britain - Ireland

using a symmetrical parabolic reflector antenna,

300.

240'

Fig. 4.8b 300. 280.

no

Fig. 4.9a

,

200

~ ~,L ....

,

"

.bo

.80

. '

~

9(lW.d.)

)15 - i'.b

_,

*

IC,-_. _ '100

,

-3dB contour of one

-120

spotbeam.

e

optimization point. - - - transformed contour of

. ILjo

Great Britain - Ireland

The normalized power distribution over Great Britain - Ireland using a symmetrical parabolic reflector antenna,

0= 4.5 m, p = 1, q = 10-0 . 5 , Flo = 0.6, freq. = 20 GHz.

o

'340

140

.bo

.80

I

_~ t}(~)

tattLb _____

. 100

120 . . - transformed contour of the Benelux.

lines of constant power.

The normalized power distribution over the Benelux using a symmetrical parabolic reflector antenna,

o

340

'32.0 300. 'ABo.

240 .

Fig. 4.9b Fig. 4.10 ,,--,

•

2ltc'

200

,

_,

4

S

. IyO

.bo

.80

• 100 • 12.0 -3dB contour of one spotbeam . • optimization point. - - - transformed contour of the Benelux.

The normalized power distribution over the Benelux using a symmetrical parabolic reflector antenna,

D

=

10 m, f(r)

=

1, F/D

=

0.67, freg.

=

20.4 GHz.

I

The geometry of a defocused offset fed parabolic reflector antenna.

Fig. 4.11 : The shifted ~',

<'

coordinate system .

. i .

4 . b .

S .

io'

\9.'

l~

.

f-+

~d)

300"

~

\

~(€l,O~ (d.~~i'

I 10-:3.

'-~'-b/

.. ' ..

///0

_,

_.

_,

!loo·

IOC' ,

...

.4

:~

Fig. 4.12 a

Fig. 4.12 The radia"ion pattern of antenna illuminated by a

°

n

=

16.448, ~o

=

35 , ~a / a : e . = I . 1 3 A ; S =e x y z Fig. b-h ex" = k

*

1.13 A (k i-l : c _'yll ~ k

*

1.13 A (k

a defocused offset parabolic reflector

feed with a cosine illumination function,

=

30°, oj).

=

300. = O. number of beamwidths number of beamwidths scanned},e: ,,=£ ,,=0 Y z scanned),e:x"=£z"=O

Fig. 4.12 b

Fig. 4.12 d

-"oJ. ..

-12. -\0

-II -b

-'1 -2 0 2.' ~

b'

Ii .

10' 1 2 ' -

r

9<1Iad.)

~I.ob, 10'3

Joe>

t

~

~

'3

<8,o)}

CdeIJ}

:106.

100·

Fig. 4.12 f

-10·

Fig. 4.12 h

-20 .

. -i:1 .

-10'"':8 .

_~

--;";''1

-_

>o-. ~2."""''1,...

b .

8

10 . lit. •

~

G

~)

-::It» - -

"'I.ebl

\.O-~

-1000

Fig. 4.12 j

-10·

-20·

-\0'